Properties of Square Numbers
The below table shows the squares of numbers from 1 to 20
| Number | Square | Number | Square | Number | Square | Number | Square |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 6 | 36 | 11 | 12 | 16 | 25 |
| 2 | 4 | 7 | 49 | 12 | 14 | 17 | 28 |
| 3 | 9 | 8 | 64 | 13 | 16 | 18 | 32 |
| 4 | 16 | 9 | 81 | 14 | 19 | 19 | 36 |
| 5 | 25 | 10 | 100 | 15 | 22 | 20 | 40 |
Study the square numbers in the above table.
What are the ending digits (that is, digits in the units place) of the square numbers from 1-10? Now, can you use that information to fill the unit digits for squares from 11 to 20 ?
All these numbers end with
None of these end with the digits
Can we say that if a number ends in 0, 1, 4, 5, 6 or 9, then it must be a square number? Let's find out.
Can we say whether the following numbers are perfect squares?
Try these
1. Write five numbers which you can decide by looking at their units digit that they are not square numbers.
For a number to be a
If a number ends in 2, 3, 7, or 8, it is
Therefore, five numbers that are not square numbers based solely on their unit digits are:
2. Write five numbers which you cannot decide just by looking at their units digit (or units place) whether they are square numbers or not.
If a number ends in 0, 1, 4, 5, 6, or 9, it might be a square number,
For example:
Now fill in the units digit in the following table of some numbers and their squares like earlier. Again, observe the one’s place in both.
| Number | Square | Number | Square | Number | Square | Number | Square |
|---|---|---|---|---|---|---|---|
| 11 | 121 | 16 | 256 | 21 | 44 | 30 | 90 |
| 12 | 144 | 17 | 289 | 22 | 48 | 35 | 122 |
| 13 | 169 | 18 | 324 | 23 | 52 | 40 | 160 |
| 14 | 196 | 19 | 361 | 24 | 57 | 45 | 202 |
| 15 | 225 | 20 | 400 | 25 | 62 | 50 | 250 |
The following square numbers end with digit 1
| Number | Square |
|---|---|
| 1 | 1 |
| 9 | 81 |
| 11 | 121 |
| 19 | 361 |
| 21 | 441 |
TRY THESE
Which of following square numbers would end with digit 1?
We see that if a number has 1 or 9 in the units place, then it’s square ends in
Let's consider square numbers ending with 6 at the unit's place:
| Number | Square |
|---|---|
| 4 | 16 |
| 6 | 36 |
| 14 | 196 |
| 16 | 256 |
Which of the following numbers would have digit 6 at unit place.
We can see that when a square number ends in 6, the number whose square it is, will have either
TRY THESE
What will be the “one’s digit” in the square of the following numbers?
Now, consider the following given numbers and their corresponding squares
| Number | Square |
|---|---|
| 100 | |
| 400 | |
| 6400 | |
| 10000 | |
| 40000 | |
| 490000 | |
| 810000 |
Now, let's make some observations:
(a) If a number contains 3 zeros at the end, how many zeros will its square have ?
(b) What do you notice about the number of zeros at the end of the number and the number of zeros at the end of its square?
(c) Can we say that square numbers can only have even number of zeros at the end?
What can you say about the squares of even numbers and squares of odd numbers?
For even numbers:
- We can write even numbers as
where n takes the values 1,2,3,...... - When we take the square we get:
- This tells us that the square will always be divisible by
and will contain the factor of . This also tells us that it will also be . - These are the general observations for a square of an even number.
For odd numbers:
- We can write odd numbers as
where n takes the values 1,2,3,...... - When we take the square we get:
- Upon expanding, we get that the square of an odd number will be
more than the multiple of and it will also be . - These are the general observations for a square of an odd number.
The square of which of the following numbers would be an odd number/an even number?